More Games of No Chance
MSRI Publications
Volume 42, 2002
Phutball Endgames are Hard
ERIK D. DEMAINE, MARTIN L. DEMAINE, AND DAVID EPPSTEIN
Abstract. We show that, in John Conway’s board game Phutball (or
Philosopher’s Football), it is NP-complete to determine whether the current
player has a move that immediately wins the game. In contrast, the similar
problems of determining whether there is an immediately winning move in
checkers, or a move that kings a man, are both solvable in polynomial time.
1. Introduction
John Conway’s game Phutball [1–3,13], also known as Philosopher’s Football,
starts with a single black stone (the ball) placed at the center intersection of a
rectangular grid such as a Go board. Two players sit on opposite sides of the
board and take turns. On each turn, a player may either place a single white
stone (a man) on any vacant intersection, or perform a sequence of jumps. To
jump, the ball must be adjacent to one or more men. It is moved in a straight line
(orthogonal or diagonal) to the first vacant intersection beyond the men, and the
men so jumped are immediately removed (Figure 1). If a jump is performed, the
same player may continue jumping as long as the ball continues to be adjacent
to at least one man, or may end the turn at any point. Jumps are not obligatory:
one can choose to place a man instead of jumping. The game is over when a
jump sequence ends on or over the edge of the board closest to the opponent (the
opponent’s goal line) at which point the player who performed the jumps wins.
It is legal for a jump sequence to step onto but not over one’s own goal line.
One of the interesting properties of Phutball is that any move could be played
by either player, the only partiality in the game being the rule for determining
the winner.
It is theoretically possible for a Phutball game to return to a previous position, so it may be necessary to add a loop-avoidance rule such as the one in
Chess (three repetitions allow a player to claim a draw) or Go (certain repeated
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ERIK D. DEMAINE, MARTIN L. DEMAINE, AND DAVID EPPSTEIN
Figure 1. A jump in Phutball. Left: The situation prior to the jump. Right: The
situation after jumping two men. The same player may then jump the remaining
man.
positions are disallowed). However, repetitions do not seem to come up much in
actual practice.
It is common in other board games1 that the problems of determining the outcome of the game (with optimal play), or testing whether a given move preserves
the correct outcome, are PSPACE-complete [5], or even EXPTIME-complete for
loopy games such as Chess [8] and Go [11]. However, no such result is known
for Phutball. Here we prove a different kind of complexity result: the problem
of determining whether a player has a move that immediately wins the game (a
mate in one, in chess terminology) is NP-complete. Such a result seems quite
unusual, since in most games there are only a small number of legal moves, which
could all be tested in polynomial time. The only similar result we are aware of
is that, in Twixt, it is NP-complete to determine whether a player’s points can
be connected to form a winning chain [10]. However, the Twixt result seems to
be less applicable to actual game play, since it depends on a player making a
confusing tangle of his own points, which does not tend to occur in practice. The
competition between both players in Phutball to form a favorable arrangement
of men seems to lead much more naturally to complicated situations not unlike
the ones occurring in our NP-completeness proof.
2. The NP-Completeness Proof
Testing for a winning jump sequence is clearly in NP, since a jump sequence
can only be as long as the number of men on the board. As is standard for
NP-completeness proofs, we prove that the problem is hard for NP by reducing
a known NP-complete problem to it. For our known problem we choose 3-SAT:
satisfiability of Boolean formulae in conjunctive normal form, with at most three
variables per clause. We must show how to translate a 3-SAT instance into a
Phutball position, in polynomial time, in such a way that the given formula is
solvable precisely if there exists a winning path in the Phutball position.
1
More precisely, since most games have a finite prescribed board size, these complexity
results apply to generalizations in which arbitrarily large boards are allowed, and in which the
complexity is measured in terms of the board size.
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Figure 2. Overall plan of the NP-completeness reduction: a path zigzags through
horizontal pairs of lines (representing variables) and vertical triples of lines (representing clauses). Certain variable-clause crossings are marked, representing an
interaction between that variable and clause.
The overall structure of our translation is depicted in Figure 2, and a small
complete example is shown in Figure 6. We form a Phutball position in which
the possible jump sequences zigzag horizontally along pairs of lines, where each
pair represents one of the variables in the given 3-SAT instance. The path then
zigzags vertically up and down along triples of lines, where each triple represents
one of the clauses in the 3-SAT instance. Thus, the potential winning paths are
formed by choosing one of the two horizontal lines in each pair (corresponding to
selecting a truth value for each variable) together with one of the three vertical
lines in each triple (corresponding to selecting which of the three variables has a
truth value that satisfies the clause). By convention, we associate paths through
the upper of a pair of horizontal lines with assignments that set the corresponding
variable to true, and paths through the lower of the pair with assignments that
set the variable to false. The horizontal and vertical lines interact at certain
marked crossings in a way that forces any path to correspond to a satisfying
truth assignment.
We now detail each of the components of this structure.
Fan-in and fan-out: At the ends of each pair or triple of lines, we need a
configuration of men that allows paths along any member of the set of lines
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ERIK D. DEMAINE, MARTIN L. DEMAINE, AND DAVID EPPSTEIN
Figure 3. Configuration of men to allow a choice between three vertical lines.
Similar configurations are used at the other end of each triple of lines, and at
each end of pairs of horizontal lines.
Figure 4. Left: configuration of men to allow horizontal and vertical lines to
cross without interacting. Right: after the horizontal jump has been taken, the
short gap in the vertical line still allows it to be traversed via a pair of jumps.
Figure 5. Left: configuration of men to allow horizontal and vertical lines to
interact. Right: after the horizontal jump has been taken, the long gap in the
vertical line prevents passage.
PHUTBALL ENDGAMES ARE HARD
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to converge, and then to diverge again at the next pair or triple. Figure 3
depicts such a configuration for the triples of vertical lines; the configuration
for the horizontal lines is similar. Note that, if a jump sequence enters the
configuration from the left, it can only exit through one of the three lines at
the bottom. If a jump sequence enters via one of the three vertical lines, it can
exit horizontally or on one of the other vertical lines. However, the possibility
of using more than one line from a group does not cause a problem: a jump
sequence that uses the second of two horizontal lines must get stuck at the
other end of the line, and a jump sequence that uses two of three vertical lines
must use all three lines and can be simplified to a sequence using only one of
the three lines.
Non-interacting line crossing: Figure 4 depicts a configuration of men that
allows two lines to cross without interacting. A jump sequence entering along
the horizontal or vertical line can and must exit along the same line, whether
or not the other line has already been jumped.
Interaction: Figure 5 depicts a configuration of men forming an interaction
between two lines. In the initial configuration, a jump sequence may follow
either the horizontal or the vertical line. However, once the horizontal line
has been jumped, it will no longer be possible to jump the vertical line.
Theorem 1. Testing whether a Phutball position allows a winning jump sequence is NP-complete.
Proof. As described above, we reduce 3-SAT to the given problem by forming a
configuration of men with two horizontal lines of men for each variable, and three
vertical lines for each clause. We connect these lines by the fan-in and fan-out
gadget depicted in Figure 3. If variable i occurs as the jth term of clause k, we
place an interaction gadget (Figure 5) at the point where the bottom horizontal
line in the ith pair of horizontal lines crosses the jth vertical line in the kth triple
of vertical lines. If instead the negation of variable i occurs in clause k, we place
an interaction gadget similarly, but on the top horizontal line in the pair. At
all other crossings of horizontal and vertical lines we place the crossing gadget
depicted in Figure 4. Finally, we form a path of men from the final fan-in gadget
(the arrow in Figure 2) to the goal line of the Phutball board.
The lines from any two adjacent interaction gadgets must be separated by
four or more units, but other crossing types allow a three-unit separation. By
choosing the order of the variables in each clause, we can make sure that the
first variable differs from the last variable of the previous clause, avoiding any
adjacencies between interaction gadgets. Thus, we can space all lines three units
apart. If the 3-SAT instance has n variables and m clauses, the resulting Phutball
board requires 6n + O(1) rows and 9m + O(1) columns, polynomial in the input
size.
Finally, we must verify that the 3-SAT instance is solvable precisely if the
Phutball instance has a winning jump sequence. Suppose first that the 3-SAT
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ERIK D. DEMAINE, MARTIN L. DEMAINE, AND DAVID EPPSTEIN
instance has a satisfying truth assignment; then we can form a jump sequence
that uses the top horizontal line for every true variable, and the bottom line
for every false variable. If a clause is satisfied by the jth of its three variables,
we choose the jth of the three vertical lines for that clause. This forms a valid
jump sequence: by the assumption that the given truth assignment satisfies
the formula, the jump sequence uses at most one of every two lines in every
interaction gadget. Conversely, suppose we have a winning jump sequence in
the Phutball instance; then as discussed above it must use one of every two
horizontal lines and one or three of every triple of vertical lines. We form a
truth assignment by setting a variable to true if its upper line is used and false
if its lower line is used. This must be a satisfying truth assignment: the vertical
line used in each clause gadget must not have had its interaction gadget crossed
horizontally, and so must correspond to a satisfying variable for the clause.
Figure 6 shows the complete reduction for a simple 3-SAT instance. We
note that the Phutball instances created by this reduction only allow orthogonal
jumps, so the rule in Phutball allowing diagonal jumps is not essential for our
result.
3. Phutball and Checkers
Phutball is similar in many ways to Checkers. As in Phutball, Checkers
players sit at opposite ends of a rectangular board, move pieces by sequences of
jumps, remove jumped pieces, and attempt to move a piece onto the side of the
board nearest the opponent. As in Phutball, the possibility of multiple jumps
allows a Checkers player to have an exponential number of available moves.
Checkers is PSPACE-complete [7] or EXPTIME-complete [12], depending on
the termination rules, but these results rely on the difficulty of game tree search
rather than the large number of moves available at any position. Does Checkers
have the same sort of single-move NP-completeness as Phutball?
It is convenient to view Checkers as being played on a nonstandard diamondshaped grid of square cells, with pieces that move horizontally and vertically,
rather than the usual pattern of diagonal moves on a checkerboard (Figure 7).
This view does not involve changing the rules of checkers nor even the geometric
positions of the pieces, only the markings of the board on which they rest. Then,
any jump preserves the parity of both the x- and y-coordinates of the jumping
piece, so at most one fourth of the board’s cells can be reached by jumps of a
given piece (Figure 8, left).
For any given piece p, form a bipartite graph Gp by connecting the vacant
positions that p can reach by jumping with the adjacent pieces of the opposite
color that p can jump. If p is a king, this graph should be undirected, but
otherwise it should be directed according to the requirement that the piece not
move backwards. Note that each jumpable piece has degree two in this graph,
PHUTBALL ENDGAMES ARE HARD
Figure 6. Complete translation of 3-SAT instance (a ∨ b ∨ c) ∧ (¬a ∨ ¬b ∨ ¬c).
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ERIK D. DEMAINE, MARTIN L. DEMAINE, AND DAVID EPPSTEIN
Figure 7. The checkerboard can equivalently be viewed as a diamond-shaped
grid of orthogonally adjacent square cells.
Figure 8. Left: only cells of the same parity can be reached by jumps. Right:
graph Gp formed by connecting jumpable pieces with cells that can be reached
by jumps from the upper black king.
so the possible sequences of jumps are simply the graph paths that begin at the
given piece and end at a vacant square. Figure 8, right, depicts an example;
note that opposing pieces that can not be jumped (because they are on cells of
the wrong parity, or because an adjacent cell is occupied) are not included in
Gp . Using this structure, it is not hard to show that Checkers moves are not
complex:
Theorem 2. For any Checkers position (on an arbitrary-size board), one can
test in polynomial time whether a checker can become a king, or whether there
is a move which wins the game by jumping all the opponent’s pieces.
Proof. Piece p can king precisely if there is a directed path in Gp from p to one
of the squares along the opponent’s side of the board. A winning move exists
precisely if there exists a piece p for which Gp includes all opposing pieces and
contains an Euler path starting at p; that is, precisely if Gp is connected and has
at most one odd-degree vertex other than the initial location of p.
PHUTBALL ENDGAMES ARE HARD
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The second claim in this theorem, testing for a one-move win, is also proved
in [7]. That paper also shows that the analogous problem for a generalization of
checkers to arbitrary graphs is NP-complete.
4. Discussion
We have shown that, in Phutball, the exponential number of jump sequences
possible in a single move, together with the ways in which parts of a jump sequence can interfere with each other, leads to the high computational complexity
of finding a winning move. In Checkers, there may be exponentially many jump
sequences, but jumps can be performed independently of each other, so finding
winning moves is easy. What about other games?
In particular, Fanorona [4,6] seems a natural candidate for study. In this game,
capturing is performed in a different way, by moving a piece in one step towards
or away from a contiguous line of the opponent’s pieces. Board squares alternate
between strong (allowing diagonal moves) and weak (allowing only orthogonal
moves), and a piece making a sequence of captures must change direction at
each step. Like Checkers (and unlike Phutball) the game is won by capturing all
the opponent’s pieces rather than by reaching some designated goal. Is finding
a winning move in Fanorona hard? If so, a natural candidate for a reduction is
the problem of finding Hamiltonian paths in grid graphs [9].
The complexity of determining the outcome of a general Phutball position
remains open. We have not even proven that this problem is NP-hard, since
even if no winning move exists in the positions we construct, the player to move
may win in more than one move.
Acknowledgment
This work was inspired by various people, including John Conway and Richard
Nowakowski, playing phutball at the MSRI Combinatorial Game Theory Research Workshop in July 2000.
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Erik D. Demaine
MIT Laboratory for Computer Science
200 Technology Square
Cambridge, MA 02139
United States
[email protected]
Martin L. Demaine
MIT Laboratory for Computer Science
200 Technology Square
Cambridge, MA 02139
United States
[email protected]
David Eppstein
University of California, Irvine
Department of Information and Computer Science
Irvine, CA 92697-3425
United States
[email protected]